Properties

Label 2-4788-1.1-c1-0-34
Degree $2$
Conductor $4788$
Sign $-1$
Analytic cond. $38.2323$
Root an. cond. $6.18323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 2·11-s + 2·13-s − 19-s − 2·23-s − 5·25-s + 6·29-s + 8·31-s − 6·37-s + 12·43-s − 8·47-s + 49-s − 14·53-s − 8·59-s − 2·61-s + 6·71-s − 2·73-s + 2·77-s + 4·79-s − 16·83-s − 16·89-s − 2·91-s − 2·97-s − 12·101-s + 18·107-s + 6·109-s − 6·113-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.603·11-s + 0.554·13-s − 0.229·19-s − 0.417·23-s − 25-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 1.92·53-s − 1.04·59-s − 0.256·61-s + 0.712·71-s − 0.234·73-s + 0.227·77-s + 0.450·79-s − 1.75·83-s − 1.69·89-s − 0.209·91-s − 0.203·97-s − 1.19·101-s + 1.74·107-s + 0.574·109-s − 0.564·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(38.2323\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4788,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
19 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.072348116622340362415541171425, −7.21305150315764483087390090425, −6.33969081927775734412645760527, −5.91559710311683439820285187631, −4.90788451211076257560752873720, −4.20320976383562817002856830656, −3.26168525541079232864382782586, −2.50490787196561638135161386003, −1.36270374918222285778714784866, 0, 1.36270374918222285778714784866, 2.50490787196561638135161386003, 3.26168525541079232864382782586, 4.20320976383562817002856830656, 4.90788451211076257560752873720, 5.91559710311683439820285187631, 6.33969081927775734412645760527, 7.21305150315764483087390090425, 8.072348116622340362415541171425

Graph of the $Z$-function along the critical line